The fixed-angle Bergman N-polynomial content conjecture for triangles

Let $N\in\mathbb N$ and let $\theta\in(0,\pi)$. Consider triangles of area $1$ having a fixed interior angle $\theta$, and let $\rho_N$ denote their Bergman $N$-polynomial content squared. Fixed-angle triangle conjecture. The triangle with area $1$ and fixed interior angle $\theta$ that maximizes $\rho_N$ is the isosceles triangle with area $1$ and interior angle $\theta$ opposite the base. This is posed as an analogue of fixed-angle extremal results for torsional rigidity; it remains unproved in the source.